Let ( h,k) be the center of the circle
Since the square of the radius = the square of the radius we can set up this equation
(h - 1)^2 + ( k - 1)^2 = (h - 1)^2 + (k - 3)^2 simplify
(k -1)^2 = (k - 3)^2
k^2 - 2k + 1 = k^2 - 6k + 9
-2k + 1 = -6k + 9
4k = 8
k = 2
Then we have that
(h -1)^2 + (k-1)^2 = ( h - 9)^2 + (k - 2)^2
Subbing in for k we have
(h -1)^2 + (2 -1)^2 = (h -9)^2 + (2-2)^2 simplify
h^2 - 2h + 1 + 1 = h^2 - 18h + 81
-2h + 2 = -18h+ 81
16h = 79
h = 79/16
So the center is (79/16 ,2)
And the radius can be found as
(79/16 - 1)^2 + (2 -1)^2 = r^2
(63/16)^2 + 1 = r^2
4225/256 = r^2
So the equation is
( x - 79/16)^2 + (y - 2)^2 = 4225/256